Question

If \[a,b,c\] are in A.P., then \[\frac{a}{{bc}}\],\[\frac{1}{c}\],\[\frac{2}{b}\] are in which of the following progression?
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hints:
To answer this question, you must be familiar with A.P. In this question, a, b, and c are all in A.P., which means that the double of the second term is the sum of the first and third terms, that is, \[2b = a + c\]. Simply convert our given equation \[\frac{a}{{bc}},\frac{1}{c},\frac{1}{b}\] into the general equation of A.P. to get the correct answer.
Formula used:
If the provided sequence is in Arithmetic Progression, it must meet the criterion 
\[2y = x + z\]
Complete Step-by-Step Solution:
The variables a, b, and c are shown to be in arithmetic progression.
We know that when variables are in arithmetic progression, the second term is the sum of the first and third terms.
The first term is a, the second is b, and the third is c.
That means, \[2b = a + c\]
Then, verifying the given terms for the type of progression they hold in between them as follows:
The given terms are:
\[\frac{a}{{bc}},\frac{1}{c},\frac{1}{b}\]
Since we know that a, b and c are in A.P., we can verify the arithmetic mean of the above terms (required terms). So, we can calculate the middle term as follows:
(Here the arithmetic mean of three terms is that the average value of the first and last terms results in the middle term)
$\begin{align}
  & =\frac{\frac{a}{bc}+\frac{2}{b}}{2} \\
 & =\frac{(a+2c)}{2bc} \\
\end{align}$
But we know that a, b and c are in A.P. and we have $2b=a+c$. So, on substituting in the above value, we get
$\begin{align}
  & =\frac{a+c+c}{2bc} \\
 & =\frac{2b+c}{2bc}\ne \frac{1}{c} \\
\end{align}$
Therefore, we can clearly see that \[\frac{a}{{bc}},\frac{1}{c},\frac{1}{b}\] are not in A.P., G.P., and H.P. So, the correct option is D.
Hence, option D is correct
Note:
Here, we need to remember that, using means we can verify the type of the series. So, by finding the respective means such as A.M., G.M., and H.M., we can compare the obtained mean with the middle term. If they are the same then we can name that sequence by its respective progression. Otherwise, we can say that they are not in any type of progression.